https://file.aiquickdraw.com/mathbot/file/1726009731168oekupjbn.jpg

Let's analyze the graph and address both questions (a) and (b):

### Part (a): \( f(1) \)
From the graph, observe the point where \( x = 1 \). At \( x = 1 \), the graph has a **hollow dot**, indicating that the function \( f(x) \) is not defined at this point. Therefore:

\[
f(1) = \text{undefined}
\]

### Part (b): \( \lim_{x \to 1} f(x) \)
To evaluate the limit \( \lim_{x \to 1} f(x) \), we need to consider the behavior of the function as \( x \) approaches 1 from both the left-hand side (as \( x \to 1^- \)) and the right-hand side (as \( x \to 1^+ \)).

- As \( x \to 1^- \) (from the left), the graph approaches the value \( 2 \).
- As \( x \to 1^+ \) (from the right), the graph also approaches the value \( 2 \).

Since the left-hand limit and right-hand limit are equal, we conclude that the two-sided limit exists, and:

\[
\lim_{x \to 1} f(x) = 2
\]

### Summary:
- \( f(1) \) is undefined.
- \( \lim_{x \to 1} f(x) = 2 \).

Would you like any more details or explanations?

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Here are five related questions to expand your understanding:
1. What is the difference between a limit and a function's value at a point?
2. How do you evaluate one-sided limits using a graph?
3. What conditions must hold for a limit to exist at a given point?
4. How can a function have a limit at a point where it is undefined?
5. What happens if the left-hand limit and right-hand limit do not match?

**Tip**: When analyzing a graph, always check for discontinuities, as they play a crucial role in determining whether the function is defined or the limit exists at specific points.

Evaluate f(1) and lim f(x) as x approaches 1 from the graph.

This problem explores function evaluation and limits using a graph. Analyze the behavior of f(x) at x = 1, determining whether f(1) is defined and finding the limit of f(x) as x approaches 1. Suitable for students in Grades 10-12 studying limits and continuity.

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